Question

Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2 + x4 = 0 and x2 + x3 + x4 = 0 X3 X4 = Find an orthonormal basis H {h1, h2, h3, h4} for R4 with the property that hy and h2 are elements of an orthonormal basis for W, where orthonormality is defined with respect to the dot product on R4 x R4.

Answer 1

42 Wa {G ERY x + x + ₂ = 0 x + x₂ + 24 =0 } 14 Lim of giren subspace are 2. en the bass's of subspace hare only a rector the basis vector of w is jef us (0, 1, 0,-1), (2, 1,2-1). NOW we finding or the normal Basis By G.m. process B = {0, 1, 0, -1) B = (2,-, 2-1) B₂ = (1, 0, 0, o) B = (0,1,0,0) A = {B. B P P } is. Bars's of pt. B = {4, L, Xa 194) is orthonosmal Basis of RA

Now finding oothe normal basis. L 11 B, 11 I want = B2 - (B-2)2, 11 || ro ۱۱۱۱ where Vo = P2 ( P2 X) &, - (B old d4 14 11X1 r = B - (PAX,) % - (18/1 leto first orthonormal Baris. BL B, X, I IB,11 (0,1,04) BB tz (0,1,0,1) 0 =B-(BW)</ 1 = (2,1,2,-1) {(2,13,1) 04:28) County Y> (34,2,-0)-[0) 1,2 (2,-1, 2-).

0 (+1 +1- ا( / (1- درارد) - Jio | = (%۰۹) - (R). | = (1/5/۹/۵) (۲/۹/۰,۰) - ( 2 ) - (R) 4. 0 ( ت ) (009) - (۱٫۰٫۰٫۰) - (0٫۵ 1,07) - 0, 0, ( = (/0/00) / //) ر ر = (/16 (1 هر ۱٫۰٫۵) (0 ,0 ,0) & Tg = 1 //) Vio (1= (0, 10, 0) (12/05 ( 1 ) ( 1 » را )) ( 4 / 6 / ( د ) - ( 01, 0, 0)